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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to use the product property of square roots to multiply square roots.

Overview

  • The Product Property of Square Roots
  • Multiplication Rule for Square Root Expressions

The product property of square roots

In our work with simplifying square root expressions, we noted that

x y = x y

Since this is an equation, we may write it as

x y = x y

To multiply two square root expressions, we use the product property of square roots.

The product property x y = x y

x y = x y

The product of the square roots is the square root of the product.

In practice, it is usually easier to simplify the square root expressions before actually performing the multiplication. To see this, consider the following product:

8 48
We can multiply these square roots in either of two ways:

Simplify then multiply.

4 · 2 16 · 3 = ( 2 2 ) ( 4 3 ) = 2 · 4 2 · 3 = 8 6

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Multiply then simplify.

8 48 = 8 · 48 = 384 = 64 · 6 = 8 6

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Notice that in the second method, the expanded term (the third expression, 384 ) may be difficult to factor into a perfect square and some other number.

Multiplication rule for square root expressions

The preceding example suggests that the following rule for multiplying two square root expressions.

Rule for multiplying square root expressions

  1. Simplify each square root expression, if necessary.
  2. Perform the multiplecation.
  3. Simplify, if necessary.

Sample set a

Find each of the following products.

3 6 = 3 · 6 = 18 = 9 · 2 = 3 2

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8 2 = 2 2 2 = 2 2 · 2 = 2 4 = 2 · 2 = 4

This product might be easier if we were to multiply first and then simplify.

8 2 = 8 · 2 = 16 = 4

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20 7 = 4 5 7 = 2 5 · 7 = 2 35

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5 a 3 27 a 5 = ( a 5 a ) ( 3 a 2 3 a ) = 3 a 3 15 a 2 = 3 a 3 · a 15 = 3 a 4 15

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( x + 2 ) 7 x 1 = ( x + 2 ) 6 ( x + 2 ) x 1 = ( x + 2 ) 3 ( x + 2 ) x 1 = ( x + 2 ) 3 ( x + 2 ) ( x 1 ) or = ( x + 2 ) 3 x 2 + x 2

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Practice set a

Find each of the following products.

x + 4 x + 3

( x + 4 ) ( x + 3 )

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8 m 5 n 20 m 2 n

4 m 3 n 10 m

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9 ( k 6 ) 3 k 2 12 k + 36

3 ( k 6 ) 2 k 6

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2 a ( 5 a 8 a 3 )

a 10 4 a 2

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32 m 5 n 8 ( 2 m n 2 10 n 7 )

8 m 3 n 2 n 8 m 2 n 5 5 m

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Exercises

x + 2 x 3

( x + 2 ) ( x 3 )

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y + 3 y 2

( y + 3 ) ( y 2 )

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x + 9 ( x + 9 ) 2

( x + 9 ) x + 9

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3 a 2 15 a 3

3 a 2 5 a

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12 ( p q ) 3 3 ( p q ) 5

6 ( p q ) 4

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15 a 2 ( b + 4 ) 4 21 a 3 ( b + 4 ) 5

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125 m 5 n 4 r 8 8 m 6 r

10 m 5 n 2 r 4 10 m r

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7 ( 2 k 1 ) 11 ( k + 1 ) 3 14 ( 2 k 1 ) 10

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2 a 4 5 a 3 2 a 7

2 a 7 5

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2 m 3 n + 1 10 m n + 3

2 m 2 n + 2 5

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75 ( a 2 ) 7 48 a 96

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8 ( a 3 a )

2 2 a 2 6 a

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y ( y 5 + 3 y 3 )

y 2 ( y + 3 )

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8 a 5 ( 2 a 6 a 11 )

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12 m 3 ( 6 m 7 3 m )

6 m 2 ( m 3 2 1 )

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5 x 4 y 3 ( 8 x y 5 7 x )

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Exercises for review

( [link] ) Factor a 4 y 4 25 w 2 .

( a 2 y 2 + 5 w ) ( a 2 y 2 5 w )

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( [link] ) Find the slope of the line that passes through the points ( 5 , 4 ) and ( 3 , 4 ) .

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( [link] ) Perform the indicated operations:

15 x 2 20 x 6 x 2 + x 12 · 8 x + 12 x 2 2 x 15 ÷ 5 x 2 + 15 x x 2 25

4 ( x + 5 ) ( x + 3 ) 2

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( [link] ) Simplify x 4 y 2 z 6 by removing the radical sign.

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( [link] ) Simplify 12 x 3 y 5 z 8 .

2 x y 2 z 4 3 x y

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Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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AMJAD
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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I'm interested in Nanotube
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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